A Cosmic Hall of Mirrors

The Poincaré dodecahedral space (left) can be described as the interior of a “sphere” made from 12 slightly curved pentagons. However, there is one big difference between this shape and a football because when one goes out from a pentagonal face, one immediately comes back inside the ball from the opposite face after a 36° rotation. Such a multiply connected space can therefore generate multiple images of the same object, such as a planet or a photon. Other such well-proportioned, spherical spaces that fit the WMAP data are the tetrahedron (middle) and octahedron (right).